Positron emission tomography (PET) is a branch of nuclear medicine in which a positron-emitting radiopharmaceutical is introduced into the body of a patient. As the radiopharmaceutical decays, positrons are generated. More specifically, each of a plurality of positrons reacts with an electron in what is known as a positron annihilation event, thereby generating a coincident pair of gamma photons which travel substantially in opposite directions along a line of coincidence. A gamma photon pair detected within a coincidence time is ordinarily recorded by the PET scanner as an annihilation event. In time of flight (“TOF”) imaging, the time within the coincidence interval at which each gamma photon in the coincident pair is detected is also measured. The time of flight information provides an indication of the location of the detected event along the line of coincidence. Data from a plurality of annihilation events is used to reconstruct or create images of the patient or object scanned, typically by using statistical (iterative) or analytical reconstruction algorithms.
FIG. 1 illustrates the transaxial and axial coordinates of an emitted positron and the measured line of response (LOR) of a 3D detector. The coordinates (xe, ye, ze) or (se, te, ze) define the emitted positron's image coordinate. The measured LOR's projection coordinate can be defined by either (s, φ, z, θ), where z=(za+zb)/2, or may include the additional dimension t for a TOF-LOR. In these types of PET imaging devices, besides the variation in the detector efficiency of an individual crystal, the detection efficiency of the overall scanner is determined by geometric factors, which in turn depend on the solid angle formed by the area, the distance of a detector crystal to the emission point, and the incident angle of the LOR into the crystal.
The solid angle is a two-dimensional angle in three-dimensional space that an object subtends at a point. Mathematically, the solid angle Ω subtended by a surface S is written as,
                              Ω          ≡                      ∫                                          ∫                S                                                                              ⁢                                                                    n                    ^                                    ·                                      ⅆ                    a                                                                    r                  2                                                                    ,                            (        1        )            where {circumflex over (n)} is a unit vector from the point, da is the differential area of a surface patch, and r is the distance from the origin to the patch. The solid angle is a measure of how large that object appears to an observer looking from that point. An object's solid angle is equal to the area of the segment of unit sphere (centered at the vertex of the angle) restricted by the object (this definition works in any dimension, including 1D and 2D). FIG. 2A illustrates the concept of the solid angle used in the detection of coincidence events of PET. The illustrated solid section in the unit circle represents the solid angle of crystal pair (i,j), or LORij, measured from point “p”. Thus, in the PET scanner shown in FIG. 1, the solid angle of the LOR depends on the emission point position, defined by (se, te, ze). The (se, te) coordinates define the emission point position in the transverse plane (left sub-figure of FIG. 1). The ze coordinate defines the position in the axial direction of the scanner. The solid angle of the LOR increases toward the edge of the transaxial field of view, i.e., at large |s|.
When two gamma rays from the emission position individually hit two crystals of the LOR, both incident angles of two gamma rays determine the incident angle of the LOR. Each incident angle of the gamma ray can be explained by the polar angle α and the azimuthal angle β, as shown in FIG. 2B. Both angles α and β depend on the relationship between the flight direction of the gamma ray and the normal of the crystal surface at which the gamma ray enters. As angle α or β increases, the incident angle of the gamma ray also increases. The incident angle of the gamma ray determines the amount of penetration of the gamma ray into the crystal, i.e., the depth-of-interaction (DOI). As a result, when there is a larger |s| or larger θ, the normal of one crystal of the LOR is changed so that the angle α or β is increased, and DOI effects are then changed. This situation is characterized as the LOR being more aligned.
Additionally, the geometric factors are also affected by crystal positions of the LOR in the detector block. When crystals of the LOR are closer to the edge of the detector block, one side face of the detector block contributes more effects to the solid angle and the incident angle than crystals of the LOR on the front face of a detector block.
A geometric correction factor used to correct raw measured data may be determined by obtaining high-count planar or rotating line data. The raw count data are first corrected for source geometry, attenuation, and individual crystal efficiency variation. Then, radial profiles along s are generated per slice as a function of ring difference (zb-za). These radial profiles are then inverted and applied directly as the geometric correction factor.
In the conventional measurement approach, a planar source is placed at the center, or a transmission line source is rotated, to measure the correction factor, as shown in FIG. 3. Data for the geometric correction factor are collected for all detector pairs by exposing them to a planar source. The correction factor is computed based on a ratio of the average coincidence counts measured for all LORs to the counts for a particular LOR. However, the measurement approach of using either the planar source or the rotated transmission line source has some limitations. First, for the normalization of a TOF-PET scanner, when using a measurement technique to capture different emission positions, the planar source has to be placed at multiple horizontal locations, or the line source has to be rotated in different radii, as shown in FIG. 4. Using multiple acquisition scans yields several major disadvantages, such as a complicated normalization scan protocol, multiple line or planar source positioning, multiple data acquisition, and much longer scan time, for the normalization of a TOF-PET scanner. Second, although a high-count normalization scan is generally performed to reduce noise, negative effects on final images can still be expected. The noise distribution in normalization data is not uniform, i.e., there is higher noise in LORs having a big ring difference (zb-za). Also, before generating the corrective factor, other corrections are required, such as intrinsic crystal efficiencies. Additional noise can be propagated from these corrections. Moreover, current PET reconstruction is theoretically preferred to include the corrective factor into the system response in order to keep the validity of the Poisson model. Any prior methods with statistical noise will make the reconstruction deviate from the Poisson model. Thus, the resultant PET images may not be optimal from a theoretical point of view.